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Metrizable compacta in the space of continuous functions with the topology of pointwise convergence

Acta Mathematica Hungarica
Volume/Issue:
Volume 117: Issue 4
DOI:
https://doi.org/10.1007/s10474-007-6105-7
Author: V. Mykhaylyuk

Abstract  

We prove that every point-finite family of nonempty functionally open sets in a topological space X has the cardinality at most an infinite cardinal κ if and only if w(X) ≦ κ for every Valdvia compact space Y

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C p(X). Correspondingly a Valdivia compact space Y has the weight at most an infinite cardinal κ if and only if every point-finite family of nonempty open sets in C p(Y) has the cardinality at most κ, that is p(C p(Y)) ≦ κ. Besides, it was proved that w(Y) = p(C p(Y)) for every linearly ordered compact Y. In particular, a Valdivia compact space or linearly ordered compact space Y is metrizable if and only if p(C p(Y)) = ℵ0. This gives answer to a question of O. Okunev and V. Tkachuk.

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